Using implicit differentiation, find (dy)/(dx). sqrt(xy)=-2+yx^(3)

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Using implicit differentiation, find  (dy)/(dx).  sqrt(xy)=-2+yx^(3)

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Identify Equation: Identify the equation that needs to be differentiated implicitly. Equation: sqrt(xy) = -2 + yx^(3)Differentiate Both Sides: Differentiate both sides of the equation with respect to x, remembering to apply the product rule to xy and yx^(3), and the chain rule to sqrt(xy). d/dx(sqrt(xy)) = d/dx(-2 + yx^(3))Apply Chain Rule: Apply the chain rule to the left side: d/dx(sqrt(u)) = (1/2)u^(-1/2) * du/dx, where u = xy. d/dx(sqrt(xy)) = (1/2)(xy)^(-1/2) * d/dx(xy)Apply Product Rule: Apply the product rule to d/dx(xy): d/dx(uv) = u'v + uv', where u = x and v = y. d/dx(xy) = d/dx(x)y + x*d/dx(y) d/dx(xy) = y + x(dy/dx)Substitute Result: Substitute the result from the product rule into the chain rule result. (1/2)(xy)^(-1/2) * (y + x(dy/dx))Differentiate Right Side: Differentiate the right side of the equation with respect to x, applying the product rule to yx^(3). d/dx(-2 + yx^(3)) = d/dx(-2) + d/dx(yx^(3)) d/dx(-2 + yx^(3)) = 0 + d/dx(yx^(3))Apply Product Rule: Apply the product rule to d/dx(yx^(3)): d/dx(uv) = u'v + uv', where u = y and v = x^(3). d/dx(yx^(3)) = d/dx(y)x^(3) + y*d/dx(x^(3)) d/dx(yx^(3)) = (dy/dx)x^(3) + y*3x^(2)Combine Differentiated Results: Combine the differentiated results and set them equal to each other. (1/2)(xy)^(-1/2) * (y + x(dy/dx)) = 0 + (dy/dx)x^(3) + y*3x^(2)Solve for dy/dx: Solve for dy/dx by isolating terms involving dy/dx on one side of the equation. (1/2)(xy)^(-1/2) * y + (1/2)(xy)^(-1/2) * x(dy/dx) = (dy/dx)x^(3) + y*3x^(2)Subtract Terms: Subtract (dy/dx)x^(3) from both sides to get all dy/dx terms on one side. (1/2)(xy)^(-1/2) * x(dy/dx) - (dy/dx)x^(3) = y*3x^(2) - (1/2)(xy)^(-1/2) * yFactor out dy/dx: Factor out dy/dx on the left side of the equation. dy/dx * [(1/2)(xy)^(-1/2) * x - x^(3)] = y*3x^(2) - (1/2)(xy)^(-1/2) * yDivide Both Sides: Divide both sides by [(1/2)(xy)^(-1/2) * x - x^(3)] to solve for dy/dx. dy/dx = [y*3x^(2) - (1/2)(xy)^(-1/2) * y] / [(1/2)(xy)^(-1/2) * x - x^(3)]

Using implicit differentiation, find $\frac{d y}{d x}$ . $\newline$ $\sqrt{x y}=-2+y x^{3}$

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Using implicit differentiation, find dydx \frac{d y}{d x} dxdy​.\newlinexy=−2+yx3 \sqrt{x y}=-2+y x^{3} xy​=−2+yx3

Using implicit differentiation, find $\frac{d y}{d x}$ . $\newline$ $\sqrt{x y}=-2+y x^{3}$